Scientiae Mathematicae Vol. 3, No. 1(2000), 107 115 107 ITERATIVE SCHEMES FOR APPROXIMATING SOLUTIONS OF ACCRETIVE OPERATORS IN BANACH SPACES SHOJI KAMIMURA AND WATARU TAKAHASHI Received December 14, 1999 Abstract. In this paper, we introduce two iterative schemes for approximating solutions of the equation 0 Av, where A is an accretive operator which satisfies the range condition. Our methods are motivated by Halpern s type iteration and Mann s type iteration. 1. Introduction Let E be a real Banach space, let A E E be an m-accretive operator and let J r =(I + ra) 1 for r>0. Our paper is concerned with iterative schemes for solvingthe equation 0 Av. One well-known method is the following: x 0 = x E, x n+1 = J rn x n, n =0, 1, 2,..., (1.1) where {r n } is a sequence of positive real numbers. The convergence of (1.1) in Hilbert spaces has been studied by Rockafellar [16], Brézis and Lions [1], Lions [7] and Pazy [11]. The results in Banach spaces have been studied by Bruck and Reich [4], Reich [12, 13, 14], Nevanlinna and Reich [9], Bruck and Passty [3] and Jungand Takahashi [6]. On the other hand, Halpern [5] and Mann [8] introduced the followingiterative schemes for approximating a fixed point of a nonexpansive mapping T of E into itself: and x n+1 = α n x +(1 α n )Tx n, n =0, 1, 2,... (1.2) x n+1 = α n x n +(1 α n )Tx n, n =0, 1, 2,..., (1.3) respectively, where x 0 = x E and {α n } is a sequence in [0, 1]. Recently, the iterative schemes (1.2) and (1.3) have been studied extensively. See, for example, Takahashi [19, 20] and the references mentioned there. In this paper, motivated by (1.1), (1.2) and (1.3), we introduce two iterative schemes to solve 0 Av: one is Halpern s type and the other is Mann s type. Our methods will be defined for accretive operators which satisfy the range condition. 2. Preliminaries Throughout this paper, we denote the set of all nonnegative integers by N. Let E be a real Banach space with norm and let E denote the dual of E. We denote the value of y E at x E by x, y. When {x n } is a sequence in E, we denote strongconvergence 1991 Mathematics Subject Classification. Primary 47H06, 47H17. Key words and phrases. Accretive operator, resolvent, iterative scheme, strong convergence, weak convergence.
108 Shoji Kamimura and Wataru Takahashi of {x n } to x E by x n x and weak convergence by x n x. The modulus of convexity of E is defined by { } x + y δ(ε) = inf 1 : x 1, y 1, x y ε 2 for every ε with 0 ε 2. A Banach space E is said to be uniformly convex if δ(ε) > 0 for every ε > 0. Further, δ satisfies that x + y ( ε )) 2 (1 r δ r for every x, y E with x r, y r and x y ε. We also know that if C is a closed convex subset of a uniformly convex Banach space E, then for each x E, there exists a unique element u = Px C with x u = inf{ x y : y C}. Such a P is called the metric projection of E onto C. Let U = {x E : x =1}. The duality mapping J from E into 2 E is defined by J(x) ={f E : x, f = x 2 = f 2 } for every x E. The norm of E is said to be uniformly Gâteaux differentiable if for each y U, the limit x + ty x lim (2.1) t 0 t is attained uniformly for x U. It is also said to be Fréchet differentiable if for each x U, the limit (2.1) is attained uniformly for y U. It is known that if the norm of E is uniformly Gâteaux differentiable, then the duality mapping J is single valued and uniformly norm to weak continuous on each bounded subset of E. A Banach space E is said to satisfy Opial s condition [10] if for any sequence {x n } E, x n yimplies lim inf x n y < lim inf x n z for all z E with z y. Let C be a closed convex subset of E. A mapping T : C C is said to be nonexpansive if Tx Ty x y for all x, y C. We denote the set of all fixed points of T by F (T ). A closed convex subset C of E is said to have the fixed point property for nonexpansive mappings if every nonexpansive mapping of a bounded closed convex subset D of C into itself has a fixed point in D. Let D be a subset of C. A mapping P of C into D is said to be sunny if P (Px+ t(x Px)) = Px whenever Px+ t(x Px) C for x C and t 0. A mapping P of C into itself is said to be a retraction if P 2 = P. We denote the closure of the convex hull of D by cod. Let I denote the identity operator on E. An operator A E E with domain D(A) = {z E : Az } and range R(A) = {Az : z D(A)} is said to be accretive if for each x i D(A) and y i Ax i, i =1, 2, there exists j J(x 1 x 2 ) such that y 1 y 2,j 0. If A is accretive, then we have x 1 x 2 x 1 x 2 + r(y 1 y 2 ) for all x i D(A),y i Ax i,i =1, 2 and r>0. An accretive operator A is said to satisfy the range condition if D(A) r>0 R(I + ra). If A is accretive, then we can define, for each r > 0, a nonexpansive single valued mapping J r : R(I + ra) D(A) byj r =(I + ra) 1. It is called the resolvent of A. We also define the Yosida approximation A r by A r =(I J r )/r. We know that A r x AJ r x for all x R(I + ra) and A r x inf{ y : y Ax} for all x D(A) R(I + ra). We also know that for an accretive operator A which satisfies the range condition, we have A 1 0=F (J r ) for all r>0. An accretive operator A is said to be m-accretive if R(I + ra) =E for all r>0.
Iterative schemes for approximating solutions of accretive operators in Banach spaces 109 In the sequel, unless otherwise stated, we assume that A E E is an accretive operator which satisfies the range condition and J r is the resolvent of A for r>0. 3. Halpern s type iterative scheme In this section, we study the strongconvergence of Halpern s type iteration. We employ the methods of Wittmann [23] and Shioji and Takahashi [17] for the proof of the following theorem. Theorem 1. Let E be a Banach space with a uniformly Gâteaux differentiable norm and let C be a nonempty closed convex subset of E such that D(A) C r>0 R(I + ra). Assume that {α n } [0, 1] and {r n } (0, ) satisfy lim α n =0, n=0 α n = and lim r n =. Letx 0 = x C and let {x n } be a sequence generated by x n+1 = α n x +(1 α n )J rn x n, n N. If A 1 0 and {J t x} converges strongly to z A 1 0 as t, then {x n } converges strongly to z A 1 0. Proof. Let y n = J rn x n and u A 1 0. Then we have x 1 u = α 0 x +(1 α 0 )y 0 u α 0 x u +(1 α 0 ) y 0 u α 0 x u +(1 α 0 ) x 0 u = x u. If x n u x u for some n N \{0}, then we can show that x n+1 u x u. Then, by induction, {x n } is bounded. Therefore {y n } is also bounded. Next we shall show that lim sup x z,j(x n z) 0. (3.1) We know (x J t x)/t AJ t x and A rn x n Ay n. Since A is accretive, we have A rn x n x J tx,j(y n J t x) 0 t and hence x J t x, J(y n J t x) t A rn x n,j(y n J t x) for all n N and t>0. Then, since A rn x n =(x n y n )/r n 0asn, we obtain lim sup x J t x, J(y n J t x) 0 (3.2) for all t>0. Since J t x z as t and the norm of E is uniformly Gâteaux differentiable, for any ε>0, there exists t 0 > 0 such that and z J t x, J(y n J t x) ε 2 x z,j(y n J t x) J(y n z) ε 2
110 Shoji Kamimura and Wataru Takahashi for all t t 0 and n N. Then we have x J t x, J(y n J t x) x z,j(y n z) x J t x, J(y n J t x) x z,j(y n J t x) + x z,j(y n J t x) x z,j(y n z) = z J t x, J(y n J t x) + x z,j(y n J t x) J(y n z) ε (3.3) for all t t 0 and n N. Therefore, from (3.2) and (3.3), we have lim sup x z,j(y n z) lim sup x J t x, J(y n J t x) + ε ε. Since ε>0 is arbitrary, we obtain lim sup x z,j(y n z) 0. (3.4) On the other hand, since x n+1 y n = α n (x y n ) 0asand the norm of E is uniformly Gâteaux differentiable, we have lim x z,j(x n+1 z) x z,j(y n z) =0. (3.5) Combining(3.4) and (3.5), we obtain (3.1). From (1 α n )(y n z) =(x n+1 z) α n (x z), we have and hence (1 α n ) 2 y n z 2 x n+1 z 2 2α n x z,j(x n+1 z) x n+1 z 2 (1 α n ) 2 y n z 2 +2α n x z,j(x n+1 z) (1 α n ) x n z 2 +2α n x z,j(x n+1 z) for all n N. By (3.1), for any ε>0, there exists m N such that x z,j(x n+1 z) ε 2 for all n m. Hence we have x n+m+1 z 2 (1 α n+m ) x n+m z 2 + α n+m ε for all n N. By induction, we obtain { } n+m n+m x n+m+1 z 2 x m z 2 (1 α i )+ 1 (1 α i ) ε i=m ( x m z 2 exp n+m i=m for all n N. Therefore, from n=0 α n =, we have lim sup α i ) + ε i=m x n z 2 = lim sup x n+m+1 z 2 ε. Since ε>0 is arbitrary, {x n } converges strongly to z. The convergence of {J t x} as t was discussed by Takahashi and Ueda [22]. See also Reich [15].
Iterative schemes for approximating solutions of accretive operators in Banach spaces 111 Theorem 2. Let E be a reflexive Banach space whose norm is uniformly Gâteaux differentiable. Suppose that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings. Let C be a nonempty closed convex subset of E such that D(A) C r>0 R(I + ra). IfA 1 0, then the strong lim t J t x exists and belongs to A 1 0 for all x C. Further, if Px = lim t J t x for each x C, then P is a sunny nonexpansive retraction of C onto A 1 0. Let C be a nonempty closed convex subset of E and let T be a nonexpansive mapping of C into itself. Then A = I T is an accretive operator which satisfies C = D(A) r>0 R(I + ra); see Takahashi [18]. Then, putting A = I T in Theorem 1, we obtain the followingresult by usingtheorem 2. Corollary 3. Let C be a nonempty closed convex subset of a Banach space E with a uniformly Gâteaux differentiable norm and let T be a nonexpansive mapping from C into itself. Assume that {α n } [0, 1] and {r n } (0, ) satisfy lim α n =0, n=0 α n = and lim r n =. Letx 0 = x C and let {x n } be a sequence generated by y n = 1 x n + r n Ty n, 1+r n 1+r n x n+1 = α n x +(1 α n )y n, n N. If F (T ) and {z t } converges strongly to z F (T ) as t 0, then {x n } converges strongly to z F (T ), where z t is a unique element of C which satisfies z t = tx +(1 t)tz t. In the case where A is an m-accretive operator, we obtain the followingresult by using Theorem 2. Corollary 4. Let E be a Banach space with a uniformly Gâteaux differentiable norm and let A E E be an m-accretive operator. Assume that {α n } [0, 1] and {r n } (0, ) satisfy lim α n =0, n=0 α n = and lim r n =. Letx 0 = x E and let {x n } be a sequence generated by x n+1 = α n x +(1 α n )J rn x n, n N. (3.6) If A 1 0 and {J t x} converges strongly to z A 1 0 as t, then {x n } converges strongly to z A 1 0. 4. Mann s type iterative scheme In this section, we prove a weak convergence theorem for Mann s type iteration. Before provingthe theorem, we need the followingtwo lemmas. Lemma 5 (Browder [2]). Let C be a closed bounded convex subset of a uniformly convex Banach space E and let T be a nonexpansive mapping of C into itself. If {x n } converges weakly to z C and {x n Tx n } converges strongly to 0, then Tz = z. Lemma 6 (Reich [14]). Let E be a uniformly convex Banach space whose norm is Fréchet differentiable norm, let C be a nonempty closed convex subset of E and let {T 0,T 1,T 2,...} be a sequence of nonexpansive mappings of C into itself such that Let x C and S n = T n T n 1 T 0 for all n N. Then the set n=0 F (T n) is nonempty. n=0 co{s mx : m n} U consists of at most one point, where U = n=0 F (T n). For the proof of Lemma 6, see Takahashi and Kim [21]. Now we can prove the following weak convergence theorem.
112 Shoji Kamimura and Wataru Takahashi Theorem 7. Let E be a uniformly convex Banach space whose norm is Fréchet differentiable or which satisfies Opial s condition and let C be a nonempty closed convex subset of E such that D(A) C r>0 R(I + ra). Assume that {α n} [0, 1] and {r n } (0, ) satisfy lim sup α n < 1 and lim inf r n > 0. Let x 0 = x C and let {x n } be a sequence generated by x n+1 = α n x n +(1 α n )J rn x n, n N. (4.1) If A 1 0, then {x n } converges weakly to an element of A 1 0. Proof. Let u be an element of A 1 0 and y n = J rn x n. Then for l = x u, the set D = {z E : z u l} is a nonempty closed bounded convex subset of E which is invariant under J s for s>0. So {x n } D is bounded. From x n+1 u = α n x n +(1 α n )y n u α n x n u +(1 α n ) y n u x n u, lim x n u exists. Without loss of generality, we may assume that lim x n u 0. Since A is accretive, we have y n u y n u + r n 2 (A r n x n 0) and hence = y n u + 1 2 (x n y n ) = x n + y n u 2 x n u { 1 δ ( )} xn y n x u ( ) xn y n (1 α n ) x n u δ x u (1 α n ){ x n u y n u } = x n u α n x n u (1 α n ) y n u x n u x n+1 u. Then, by lim sup α n < 1 and lim x n u 0, we obtain δ( x n y n / x u ) 0. This implies x n y n 0. So, from y n J 1 y n = (I J 1 )y n = A 1 y n inf{ z : z Ay n } A rn x n = x n y n r n and lim inf r n > 0, we have y n J 1 y n 0. Further, letting v E be a weak subsequential limit of {x n } such that x ni v,wegety ni v. Then it follows from Lemma 5 that v F (J 1 )=A 1 0. We assume that E has a Fréchet differentiable norm. Putting T n = α n I +(1 α n )J rn and S n = T n T n 1 T 0, we have n=0 F (T n)=a 1 0 and {v} = n=0 co{x m : m n} A 1 0 by Lemma 6. Therefore {x n } converges weakly to an element of A 1 0. Next we assume that E satisfies Opial s condition. Let v 1 and v 2 be two weak subsequential limits of the sequence {x n } such that x ni v 1 and x nj v 2. As above, we have
Iterative schemes for approximating solutions of accretive operators in Banach spaces 113 v 1,v 2 A 1 0. We claim that v 1 = v 2. If not, we have lim x n v 1 = lim x ni v 1 < lim x ni v 2 = lim x n v 2 i i = lim x n j v 2 < lim x n j v 1 = lim x n v 1. j j This is a contradiction. Hence we have v 1 = v 2. This implies that {x n } converges weakly to an element of A 1 0. We can also study the strongconvergence of (4.1) by usingthe metric projection. Proposition 8. Let E be a uniformly convex Banach space and let C be a nonempty closed convex subset of E such that D(A) C r>0 R(I + ra). Assume {α n} [0, 1] and {r n } (0, ). Letx 0 = x C and let {x n } be a sequence generated by (4.1). IfA 1 0 and P is the metric projection of E onto A 1 0, then {Px n } converges strongly to an element of A 1 0. Proof. We have Px n+1 x n+1 Px n x n+1 α n Px n x n +(1 α n ) Px n J rn x n Px n x n for all n N. Then lim Px n x n exists. We shall show that {Px n } is a Cauchy sequence. Let a = lim Px n x n. If a = 0, then for any ε>0, there exists N 0 N such that Px n x n ε/4 for all n N 0.Ifn, m N 0, then we have Px n Px m Px n x n + x n Px N0 + Px N0 x m + x m Px m Px n x n + x N0 Px N0 + Px N0 x N0 + x m Px m ε. Then {Px n } is Cauchy. Let a>0. If {Px n } is not Cauchy, then there exists ε>0 such that for any N N, there are n, m N with Px n Px m ε. Choose d>0 such that ( ) a ε a + d > 1 δ a + d and N 1 N such that Px n x n <a+ d for all n N 1. For this N 1 N, there exist n, m N 1 such that Px n Px m ε. For all l N with l n and l m, we have Px n x l Px n x n <a+ d and Px m x l Px m x m <a+ d. Since E is uniformly convex, we obtain a = lim Px l x l l lim sup Px n + Px m x l l 2 ( ( )) ε (a + d) 1 δ a + d <a,
114 Shoji Kamimura and Wataru Takahashi which is a contradiction. Thus the proof is completed. As direct consequences of Theorem 7, we obtain the followingtwo results. Corollary 9. Let C be a nonempty closed convex subset of a uniformly convex Banach space E whose norm is Fréchet differentiable or which satisfies Opial s condition and let T be a nonexpansive mapping of C into itself. Assume that {α n } [0, 1] and {r n } (0, ) satisfy lim sup α n < 1 and lim inf r n > 0. Let x 0 = x C and let {x n } be a sequence generated by y n = 1 x n + r n Ty n, 1+r n 1+r n x n+1 = α n x n +(1 α n )y n, n N. If F (T ), then {x n } converges weakly to an element of F (T ). Corollary 10. Let E be a uniformly convex Banach space whose norm is Fréchet differentiable or which satisfies Opial s condition and let A E E be an m-accretive operator. Assume that {α n } [0, 1] and {r n } (0, ) satisfy lim sup α n < 1 and lim inf r n > 0. Letx 0 = x E and let {x n } be a sequence generated by x n+1 = α n x n +(1 α n )J rn x n, n N. If A 1 0, then {x n } converges weakly to an element of A 1 0. References [1] H. Brézis and P. L. Lions, Produits infinis de resolvants, Israel J. Math. 29 (1978), 329 345. [2] F. E. Browder, Semicontractive and semiaccretie nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc. 74 (1968), 660 665. [3] R. E. Bruck and G. B. Passty, Almost convergence of the infinite product of resolvents in Banach spaces, Nonlinear Anal. 3 (1979), 279 282. [4] R. E. Bruck and S. Reich, Nonexpansive projections and resolvents of accretive operators in Banach spaces, Houston J. Math. 3 (1977), 459 470. [5] B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967), 957 961. [6] J. S. Jung and W. Takahashi, Dual convergence theorems for the infinite products of resolvents in Banach spaces, Kodai Math. J. 14 (1991), 358 364. [7] P. L. Lions, Une methode iterative de resolution d une inequation variationnelle, Israel J. Math. 31 (1978), 204 208. [8] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506 510. [9] O. Nevanlinna and S. Reich, Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces, Israel J. Math. 32 (1979), 44 58. [10] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591 597. [11] A. Pazy, Remarks on nonlinear ergodic theory in Hilbert space, Nonlinear Anal. 6 (1979), 863 871. [12] S. Reich, On infinite products of resolvents, Atti Acad. Naz. Lincei 63 (1977), 338 340. [13], Constructive techniques for accretive and monotone operators, Applied Nonlinear Analysis, Academic Press, New York, 1979, pp. 335 345. [14], Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67 (1979), 274 276. [15], Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980), 287 292. [16] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control and Optim. 14 (1976), 877 898. [17] N. Shioji and W. Takahashi, Strong convergence theorems of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (1997), 3641 3645. [18] W. Takahashi, Nonlinear Functional Analysis, Kindai-kagaku-sha, Tokyo, 1988 (Japanese). [19], Fixed point theorems and nonlinear ergodic theorems for nonlinear semigroups and their applications, Nonlinear Anal. 30 (1997), 1283 1293.
Iterative schemes for approximating solutions of accretive operators in Banach spaces 115 [20], Fixed point theorems, convergence theorems and their applications, Nonlinear Analysis and Convex Analysis (W. Takahashi and T. Tanaka, eds.), World Scientific, 1999, pp. 87 94. [21] W. Takahashi and G. E. Kim, Approximating fixed points of nonexpansive mappings in Banach spaces, Math. Japon. 48 (1998), 1 9. [22] W. Takahashi and Y. Ueda, On Reich s strong convergence theorems for resolvents of accretive operators, J. Math. Anal. Appl. 104 (1984), 546 553. [23] R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992), 486 491. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Ohokayama, Meguro-ku, Tokyo 152-8552, Japan E-mail address, S. Kamimura: kamimura@is.titech.ac.jp E-mail address, W. Takahashi: wataru@is.titech.ac.jp